[Paper Review] Vanishing of Ext and Tor over Cohen-Macaulay local rings

Craig Huneke, Liana M. Şega+1 more|arXiv (Cornell University)|Sep 3, 2014

Commutative Algebra and Its Applications12 references3 citations

TL;DR

This paper establishes the Auslander-Reiten conjecture for commutative Artinian local rings with ${\mathfrak{m}}^3 = 0$, proving that if $\operatorname{Ext}^i_R(M, M \oplus R) = 0$ for four consecutive $i \geq 2$, then $M$ is free. It further shows that for Gorenstein rings in this class, vanishing of $\operatorname{Ext}^i_R(M,M)$ for a single $i > 0$ implies freeness, and provides bounds on the number of vanishing Ext modules required, linking the vanishing of Tor and Ext to the vanishing of $\mathfrak{m}^3$. The results are derived via duality and Poincaré series analysis, with implications for the Tachikawa conjecture in low type rings.

ABSTRACT

We consider vanishing of Ext and Tor, especially over Artinian rings. In particular, we prove the Auslander-Reiten conjecture for all commutative local rings in which the cube of the maximal ideal is zero.

Motivation & Objective

• To prove the Auslander-Reiten conjecture for commutative Artinian local rings with $\mathfrak{m}^3 = 0$, where $\operatorname{Ext}^i_R(M, M \oplus R) = 0$ for four consecutive $i \geq 2$ implies $M$ is free.
• To extend the result to Gorenstein rings by showing that $\operatorname{Ext}^i_R(M,M) = 0$ for a single $i > 0$ implies $M$ is free.
• To establish effective bounds on the number of vanishing Ext modules required for freeness, depending on $\nu(M)$ and $\nu(\mathfrak{m}M)$, when $\mathfrak{m}^2M = 0$.
• To investigate the relationship between the vanishing of $\operatorname{Tor}_i^R(M,N)$ and the nilpotency of the maximal ideal, proposing a conjecture that $\mathfrak{m}^{p+q-1} = 0$ if $\operatorname{Tor}_i^R(M,N) = 0$ for all $i > 0$ and $\mathfrak{m}^pM = \mathfrak{m}^qN = 0$.
• To prove the Tachikawa conjecture for Cohen-Macaulay rings of type at most 2, showing that $\operatorname{Tor}_2^R(\omega,\omega) = 0$ or $\operatorname{Ext}^i_R(\omega,R) = 0$ for $i=1,2$ implies $R$ is Gorenstein.

Proposed method

• Use of Matlis duality to translate vanishing of $\operatorname{Ext}^i_R(M,N)$ into vanishing of $\operatorname{Tor}_i^R(M,N)$, enabling homological analysis via Tor modules.
• Application of Poincaré series and Hilbert series to analyze the structure of $R$-modules, particularly when $\mathfrak{m}^3 = 0$ or $\mathfrak{m}^2M = 0$.
• Employment of the Yoneda product structure on $\operatorname{Ext}_R^*(k,k)$ to show that vanishing of certain Tor modules implies the $k$-algebra $\operatorname{Ext}_R^*(k,k)$ is generated in degree 1, hence $R$ is Koszul.
• Use of exterior powers $\Lambda^n_R(\varphi)$ and the induced map $\varphi \otimes_R \varphi$ to analyze the rank and structure of syzygies, particularly in short exact sequences involving $R^2$.
• Leveraging Nakayama’s Lemma and the additivity of multiplicity to show that a module $N$ with $\operatorname{Tor}_2^R(M,M) = 0$ and $M$ faithful must be cyclic, leading to a contradiction if $R$ is not Gorenstein.
• Reduction of the Tachikawa conjecture to the case of $\mathfrak{m}^3 = 0$ via syzygy replacement and duality, followed by analysis of the canonical module $\omega$ in low-type rings.

Experimental results

Research questions

• RQ1 Does the vanishing of $\operatorname{Ext}^i_R(M, M \oplus R)$ for four consecutive $i \geq 2$ imply that $M$ is free over a commutative Artinian local ring with $\mathfrak{m}^3 = 0$?
• RQ2 If $R$ is Gorenstein and $\operatorname{Ext}^i_R(M,M) = 0$ for some $i > 0$, is $M$ necessarily free?
• RQ3 Can the number of vanishing $\operatorname{Ext}^i_R(M,M \oplus R)$ modules be bounded in terms of $\nu(M)$ and $\nu(\mathfrak{m}M)$ when $\mathfrak{m}^2M = 0$?
• RQ4 If $\operatorname{Tor}_i^R(M,N) = 0$ for all $i > 0$ and $\mathfrak{m}^pM = \mathfrak{m}^qN = 0$, does it follow that $\mathfrak{m}^{p+q-1} = 0$?
• RQ5 Does the Tachikawa conjecture hold for Cohen-Macaulay local rings of type at most 2, i.e., if $\operatorname{Tor}_2^R(\omega,\omega) = 0$, is $R$ Gorenstein?

Key findings

• For a commutative Artinian local ring with $\mathfrak{m}^3 = 0$, if $\operatorname{Ext}^i_R(M, M \oplus R) = 0$ for four consecutive $i \geq 2$, then $M$ is free.
• If $R$ is Gorenstein and $\operatorname{Ext}^i_R(M,M) = 0$ for some $i > 0$, then $M$ is free, extending a result of Hoshino to the commutative setting.
• When $\mathfrak{m}^2M = 0$, the vanishing of $\operatorname{Ext}^i_R(M, M \oplus R)$ for all $i$ with $0 < i \leq \max\{3, \nu(M), \nu(\mathfrak{m}M)\}$ implies $M$ is free.
• The conjecture that $\mathfrak{m}^{p+q-1} = 0$ follows from $\operatorname{Tor}_i^R(M,N) = 0$ for all $i > 0$ and $\mathfrak{m}^pM = \mathfrak{m}^qN = 0$ is proven in the case of standard graded rings.
• For Cohen-Macaulay local rings of type at most 2, $\operatorname{Tor}_2^R(\omega,\omega) = 0$ implies $R$ is Gorenstein, and $\operatorname{Ext}^i_R(\omega,R) = 0$ for $i=1,2$ also implies $R$ is Gorenstein.
• The vanishing of $\operatorname{Tor}_i^R(k, \mu_{\mathfrak{m}})$ for all $i$ implies $\operatorname{Ext}_R^*(k,k)$ is generated in degree 1, so $R$ is Koszul, and hence $\mathfrak{m}^3 = 0$.